Tamarkin Category: Microlocal Sheaf Invariants

• Tamarkin Category is a monoidal, triangulated category defined as a quotient of the derived category of sheaves on M×ℝ, encoding microlocal and symplectic invariants.
• It admits a convolution product and rich internal structure that supports quantitative invariants such as interleaving metrics and connections to Novikov rings.
• Recent developments relate the Tamarkin Category to persistence modules, derived complete Novikov modules, and Fukaya-type categories, unifying sheaf-theoretic and symplectic frameworks.

The Tamarkin category is a monoidal, triangulated (often stable ∞-) category constructed as a quotient of the derived category of sheaves on a product manifold with $\mathbb R$. It encodes a sheaf-theoretic approach to symplectic topology, providing a microlocal model for symplectic invariants, persistence, and quantization. The Tamarkin category admits a convolution product, rich internal structure, and critical connections to Novikov rings, persistence modules, and Fukaya-type categories. Core to its utility are its support conditions, relation to microsupport, and its quantitative enhancement via interleaving metrics. Recent research establishes “almost equivalence” with categories of (derived) complete Novikov modules, and positions the Tamarkin category as a central object unifying sheaf-theoretic, persistence-theoretic, and symplectic invariants.

1. Definition and Fundamental Construction

Let $M$ be a smooth or real analytic manifold, and $t$ the coordinate on $\mathbb R_t$. The Tamarkin category is typically realized as a Verdier quotient or left-orthogonal complement in the derived (dg or stable ∞-) category of sheaves $Sh(M\times\mathbb R_t)$: $$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$ where $\tau$ is the dual cotangent variable to $t$, and $Sh_{\{\tau\le 0\}}$ denotes those sheaves with microsupport in $\{\tau\le 0\}$ (Kuwagaki, 2024, Zhang, 2018, Tsygan, 2015). The objects of $M$0 are thus represented by sheaves on $M$1 whose nontrivial microsupport lies in $M$2.

The category is equipped with a convolution (or “star”) product along $M$3: $M$4 with $M$5 and $M$6, giving $M$7 a monoidal structure (Ike et al., 2023, Scholze, 30 May 2025, Zhang, 2018).

The reduced microsupport of $M$8, given as the image of its full microsupport intersected with $M$9 and projected via $t$0, defines a geometric correspondence with subsets of $t$1 (Asano et al., 2023, Ike et al., 2023).

2. Category-Theoretic and Algebraic Structure

The Tamarkin category, in both its non-equivariant and $t$2-equivariant forms, admits notable algebraic enhancements:

• Equivariant version: For a discrete subgroup $t$3, consider $t$4, the $t$5-equivariant derived category, and $t$6 its subcategory with $t$7 support. The $t$8-equivariant Tamarkin category is

$t$9

and is monoidal and $\mathbb R_t$0-graded (Kuwagaki, 2024, Kuwagaki et al., 20 Mar 2025).

• Module and monoidal structures: The monoidal unit is the direct sum of step sheaves $\mathbb R_t$1. Relevant endomorphism algebras recover Novikov-type semigroup or completion rings (Kuwagaki, 2024).

The Tamarkin category can be equivalently described in the language of filtered sheaves, persistence modules, and almost modules over Novikov rings. The APT (Almost–Persistence–Tamarkin) correspondence framework expresses equivalences between categories of persistence modules, filtered or $\mathbb R_t$2-microsupported sheaves, and derived-complete modules over Novikov rings (Kuwagaki et al., 20 Mar 2025).

3. Quantitative and Microlocal Invariants

The Tamarkin category supports a suite of quantitative invariants and categorical metrics central to symplectic topology:

• Interleaving, isomorphism, and weak-isomorphism distances: Given objects $\mathbb R_t$3 in $\mathbb R_t$4, the $\mathbb R_t$5-interleaving and $\mathbb R_t$6-isomorphisms are defined in terms of time-shifts and commutative diagrams involving natural morphisms. The resulting pseudo-metrics—interleaving distance ($\mathbb R_t$7), weak-isomorphism distance ($\mathbb R_t$8), and isomorphism distance ($\mathbb R_t$9)—satisfy $Sh(M\times\mathbb R_t)$0 (Asano et al., 2023, Zhang, 2018).
• Sheaf capacities: For each $Sh(M\times\mathbb R_t)$1, the sheaf-theoretic capacity $Sh(M\times\mathbb R_t)$2 is the minimal shift making $Sh(M\times\mathbb R_t)$3 $Sh(M\times\mathbb R_t)$4-torsion under the convolution. This recovers symplectic capacity-like invariants of domains in $Sh(M\times\mathbb R_t)$5 (Zhang, 2018).
• Persistence-theoretic interpretation: There is a categorical equivalence between certain constructible sheaf categories with microsupport $Sh(M\times\mathbb R_t)$6 and finite-type persistence modules, where the Tamarkin interleaving distance matches the standard barcode metric (Zhang, 2018, Kuwagaki et al., 20 Mar 2025).

4. Connections to Novikov Rings and Almost-Equivalence

A fundamental structural result is the almost equivalence of the equivariant Tamarkin category $Sh(M\times\mathbb R_t)$7 and the category of derived complete modules over the Novikov ring $Sh(M\times\mathbb R_t)$8:

• Novikov ring: For $Sh(M\times\mathbb R_t)$9, the Novikov ring is defined as

$$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$0

and modules are completed with respect to the natural filtration (Kuwagaki, 2024, Ike et al., 2023).

• Almost-equivalence theorem: The Yoneda–Morita functor from $$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$1 to $$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$2 (derived complete modules) is an almost equivalence: for any $$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$3, the global category satisfies

$$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$4

is an almost embedding, with kernel and cokernel almost zero in the sense of Gabber–Ramero (Kuwagaki, 2024, Kuwagaki et al., 20 Mar 2025).

Variants for higher-dimensional cones (“$$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$5-Tamarkin categories”) and with various coefficient or symmetry groups are available, adapting to persistent, toric, or log Calabi–Yau settings (Kuwagaki et al., 20 Mar 2025, Ike et al., 2023).

5. Symplectic Topology, Fukaya Categories, and Mirror Correspondences

The Tamarkin category serves as a sheaf-theoretic model for key symplectic and homological constructions:

• Sheaf quantization: Smooth (exact) Lagrangian branes in $$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$6 admit canonical sheaf quantizations $$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$7 with reduced microsupport $$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$8, and conversely, simple objects with prescribed microsupport quantize admissible branes (Asano et al., 2023, Ike et al., 2023).
• Separation and non-displaceability: The separation theorem states that for closed (possibly non-compact, end-conic) $$\mathcal{T}(M) := Sh(M\times\mathbb R_t) / Sh_{\{\tau\le 0\}}(M\times\mathbb R_t),$$9 with compactly disjoint projections, any $\tau$0 and $\tau$1 satisfy $\tau$2. Non-displaceability and Lagrangian intersection bounds—such as via shadow distance and interleaving metrics—follow as immediate corollaries (Ike et al., 2024, Asano et al., 2023, Zhang, 2018).
• Hamiltonian stability and energy: Under Hamiltonian isotopy, objects in $\tau$3 have distances controlled by oscillation norms, quantifying symplectic displacement energy at the categorical level (Asano et al., 2023).
• Fukaya category conjecture: The Novikov-linear (enhanced) Tamarkin category for a Liouville or Weinstein $\tau$4 admits an almost fully faithful embedding of the wrapped (or infinitesimally wrapped) Fukaya category over $\tau$5, recovering exact calculations and conjectured equivalences in mirror symmetry (Ike et al., 2023).

Hochschild cohomology of the Tamarkin category over suitable open domains is canonically isomorphic, with action filtration, to filtered symplectic cohomology (Kuo et al., 2023).

6. Enhanced Sheaves, Universality, and Higher Algebraic Perspective

The universal property of the Tamarkin category as a “coefficients” enhancement—allowing for a genuine exponential local system and Fourier transform in the Betti sheaf theory—positions it as the universal monoidal sheaf theory with these symmetries. Structures such as the category of wild or enhanced Betti sheaves ($\tau$6), where $\tau$7 denotes continuously complete $\tau$8-filtered spectra, are shown to be canonical recipients of monoidal functors from the Tamarkin category (Scholze, 30 May 2025).

From the $\tau$9-categorical viewpoint, the Tamarkin category connects fundamentally with persistence modules over $t$0, categories of almost Novikov modules, and log-perfectoid sheaf models for toric and Calabi–Yau geometries. These connections are mediated by explicit functorial correspondences, completions, and the action of $t$1 or its subgroups (Kuwagaki et al., 20 Mar 2025, Kuwagaki, 2024).

7. Examples, Calculations, and Applications

• Euclidean ball: The explicit calculation of sheaf projectors for $t$2 yields capacity and cohomology calculations matching symplectic invariants (Kuo et al., 2023, Zhang, 2018).
• Persistent homology: Correspondence between barcode decompositions in persistence theory and interleaving structures in Tamarkin categories is established by concrete calculations (Zhang, 2018, Kuwagaki et al., 20 Mar 2025).
• Toric and Landau–Ginzburg models: Sheaf-theoretic models via Tamarkin-type categories provide mirror correspondences for (perfectoid) toric varieties with Novikov coefficients (Kuwagaki et al., 20 Mar 2025).
• Lagrangian cobordisms: Sheaf quantizations of cobordisms (iterated cones, shadow distance) yield fine invariants for Lagrangian intersection and energy cost in symplectic field theory (Asano et al., 2023).

The Tamarkin category thus serves as a geometric and algebraic framework unifying microlocal sheaf theory, persistence, and symplectic/homological invariants, with deep connections to almost mathematics, mirror symmetry, and higher-categorical infrastructure.